Theorem lcfrlem35 35191

Description: Lemma for lcfr 35199. (Contributed by NM, 2-Mar-2015.)

Hypotheses

Ref	Expression
lcfrlem17.h	|- H = ( LHyp ` K )
lcfrlem17.o	|- ._|_ = ( ( ocH ` K ) ` W )
lcfrlem17.u	|- U = ( ( DVecH ` K ) ` W )
lcfrlem17.v	|- V = ( Base ` U )
lcfrlem17.p	|- .+ = ( +g ` U )
lcfrlem17.z	|- .0. = ( 0g ` U )
lcfrlem17.n	|- N = ( LSpan ` U )
lcfrlem17.a	|- A = (LSAtoms ` U )
lcfrlem17.k	|- ( ph -> ( K e. HL /\ W e. H ) )
lcfrlem17.x	|- ( ph -> X e. ( V \ { .0. } ) )
lcfrlem17.y	|- ( ph -> Y e. ( V \ { .0. } ) )
lcfrlem17.ne	|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
lcfrlem22.b	|- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )
lcfrlem24.t	|- .x. = ( .s ` U )
lcfrlem24.s	|- S = (Scalar ` U )
lcfrlem24.q	|- Q = ( 0g ` S )
lcfrlem24.r	|- R = ( Base ` S )
lcfrlem24.j	|- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
lcfrlem24.ib	|- ( ph -> I e. B )
lcfrlem24.l	|- L = (LKer ` U )
lcfrlem25.d	|- D = (LDual ` U )
lcfrlem28.jn	|- ( ph -> ( ( J ` Y ) ` I ) =/= Q )
lcfrlem29.i	|- F = ( invr ` S )
lcfrlem30.m	|- .- = ( -g ` D )
lcfrlem30.c	|- C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )

Assertion

Ref	Expression
lcfrlem35	|- ( ph -> ( ._|_ ` { ( X .+ Y ) } ) = ( L ` C ) )

Distinct variable groups:   v, k, w, x, ._|_   .+ , k, v, w, x   R, k, v, x   S, k   .x. , k, v, w, x   v, V, x   k, X, v, w, x   k, Y, v, w, x   x, .0.

Allowed substitution hints:   ph( x, w, v, k)   A( x, w, v, k)   B( x, w, v, k)   C( x, w, v, k)   D( x, w, v, k)   Q( x, w, v, k)   R( w)   S( x, w, v)   U( x, w, v, k)   F( x, w, v, k)   H( x, w, v, k)   I( x, w, v, k)   J( x, w, v, k)   K( x, w, v, k)   L( x, w, v, k)   .- ( x, w, v, k)   N( x, w, v, k)   V( w, k)   W( x, w, v, k)   .0. ( w, v, k)

Proof of Theorem lcfrlem35

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcfrlem17.h	. . . 4 |- H = ( LHyp ` K )
2		lcfrlem17.o	. . . 4 |- ._|_ = ( ( ocH ` K ) ` W )
3		lcfrlem17.u	. . . 4 |- U = ( ( DVecH ` K ) ` W )
4		lcfrlem17.v	. . . 4 |- V = ( Base ` U )
5		lcfrlem17.p	. . . 4 |- .+ = ( +g ` U )
6		lcfrlem17.z	. . . 4 |- .0. = ( 0g ` U )
7		lcfrlem17.n	. . . 4 |- N = ( LSpan ` U )
8		lcfrlem17.a	. . . 4 |- A = (LSAtoms ` U )
9		lcfrlem17.k	. . . 4 |- ( ph -> ( K e. HL /\ W e. H ) )
10		lcfrlem17.x	. . . 4 |- ( ph -> X e. ( V \ { .0. } ) )
11		lcfrlem17.y	. . . 4 |- ( ph -> Y e. ( V \ { .0. } ) )
12		lcfrlem17.ne	. . . 4 |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
13		lcfrlem22.b	. . . 4 |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )
14		eqid 2462	. . . 4 |- ( LSSum ` U ) = ( LSSum ` U )
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	lcfrlem23 35179	. . 3 |- ( ph -> ( ( ._|_ ` { X , Y } ) ( LSSum ` U ) B ) = ( ._|_ ` { ( X .+ Y ) } ) )
16		lcfrlem24.t	. . . . . 6 |- .x. = ( .s ` U )
17		lcfrlem24.s	. . . . . 6 |- S = (Scalar ` U )
18		lcfrlem24.q	. . . . . 6 |- Q = ( 0g ` S )
19		lcfrlem24.r	. . . . . 6 |- R = ( Base ` S )
20		lcfrlem24.j	. . . . . 6 |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
21		lcfrlem24.ib	. . . . . 6 |- ( ph -> I e. B )
22		lcfrlem24.l	. . . . . 6 |- L = (LKer ` U )
23	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22	lcfrlem24 35180	. . . . 5 |- ( ph -> ( ._|_ ` { X , Y } ) = ( ( L ` ( J ` X ) ) i^i ( L ` ( J ` Y ) ) ) )
24		eqid 2462	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
25		lcfrlem29.i	. . . . . 6 |- F = ( invr ` S )
26		eqid 2462	. . . . . 6 |- (LFnl ` U ) = (LFnl ` U )
27		lcfrlem25.d	. . . . . 6 |- D = (LDual ` U )
28		eqid 2462	. . . . . 6 |- ( .s ` D ) = ( .s ` D )
29		lcfrlem30.m	. . . . . 6 |- .- = ( -g ` D )
30	1, 3, 9	dvhlvec 34723	. . . . . 6 |- ( ph -> U e. LVec )
31		eqid 2462	. . . . . . 7 |- ( 0g ` D ) = ( 0g ` D )
32		eqid 2462	. . . . . . 7 |- { f e. (LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) } = { f e. (LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
33	1, 2, 3, 4, 5, 16, 17, 19, 6, 26, 22, 27, 31, 32, 20, 9, 10	lcfrlem10 35166	. . . . . 6 |- ( ph -> ( J ` X ) e. (LFnl ` U ) )
34	1, 2, 3, 4, 5, 16, 17, 19, 6, 26, 22, 27, 31, 32, 20, 9, 11	lcfrlem10 35166	. . . . . 6 |- ( ph -> ( J ` Y ) e. (LFnl ` U ) )
35		eqid 2462	. . . . . . . 8 |- ( LSubSp ` U ) = ( LSubSp ` U )
36	1, 3, 9	dvhlmod 34724	. . . . . . . 8 |- ( ph -> U e. LMod )
37	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	lcfrlem22 35178	. . . . . . . 8 |- ( ph -> B e. A )
38	35, 8, 36, 37	lsatlssel 32609	. . . . . . 7 |- ( ph -> B e. ( LSubSp ` U ) )
39	4, 35	lssel 18216	. . . . . . 7 |- ( ( B e. ( LSubSp ` U ) /\ I e. B ) -> I e. V )
40	38, 21, 39	syl2anc 671	. . . . . 6 |- ( ph -> I e. V )
41		lcfrlem28.jn	. . . . . 6 |- ( ph -> ( ( J ` Y ) ` I ) =/= Q )
42		lcfrlem30.c	. . . . . 6 |- C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )
43	4, 17, 24, 18, 25, 26, 27, 28, 29, 30, 33, 34, 40, 41, 42, 22	lcfrlem2 35157	. . . . 5 |- ( ph -> ( ( L ` ( J ` X ) ) i^i ( L ` ( J ` Y ) ) ) C_ ( L ` C ) )
44	23, 43	eqsstrd 3478	. . . 4 |- ( ph -> ( ._|_ ` { X , Y } ) C_ ( L ` C ) )
45	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 27, 41	lcfrlem28 35184	. . . . . 6 |- ( ph -> I =/= .0. )
46	6, 7, 8, 30, 37, 21, 45	lsatel 32617	. . . . 5 |- ( ph -> B = ( N ` { I } ) )
47	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 27, 41, 25, 29, 42	lcfrlem30 35186	. . . . . . 7 |- ( ph -> C e. (LFnl ` U ) )
48	26, 22, 35	lkrlss 32707	. . . . . . 7 |- ( ( U e. LMod /\ C e. (LFnl ` U ) ) -> ( L ` C ) e. ( LSubSp ` U ) )
49	36, 47, 48	syl2anc 671	. . . . . 6 |- ( ph -> ( L ` C ) e. ( LSubSp ` U ) )
50	4, 17, 24, 18, 25, 26, 27, 28, 29, 30, 33, 34, 40, 41, 42, 22	lcfrlem3 35158	. . . . . 6 |- ( ph -> I e. ( L ` C ) )
51	35, 7, 36, 49, 50	lspsnel5a 18274	. . . . 5 |- ( ph -> ( N ` { I } ) C_ ( L ` C ) )
52	46, 51	eqsstrd 3478	. . . 4 |- ( ph -> B C_ ( L ` C ) )
53	35	lsssssubg 18236	. . . . . . 7 |- ( U e. LMod -> ( LSubSp ` U ) C_ (SubGrp ` U ) )
54	36, 53	syl 17	. . . . . 6 |- ( ph -> ( LSubSp ` U ) C_ (SubGrp ` U ) )
55	10	eldifad 3428	. . . . . . . 8 |- ( ph -> X e. V )
56	11	eldifad 3428	. . . . . . . 8 |- ( ph -> Y e. V )
57		prssi 4141	. . . . . . . 8 |- ( ( X e. V /\ Y e. V ) -> { X , Y } C_ V )
58	55, 56, 57	syl2anc 671	. . . . . . 7 |- ( ph -> { X , Y } C_ V )
59	1, 3, 4, 35, 2	dochlss 34968	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ { X , Y } C_ V ) -> ( ._|_ ` { X , Y } ) e. ( LSubSp ` U ) )
60	9, 58, 59	syl2anc 671	. . . . . 6 |- ( ph -> ( ._|_ ` { X , Y } ) e. ( LSubSp ` U ) )
61	54, 60	sseldd 3445	. . . . 5 |- ( ph -> ( ._|_ ` { X , Y } ) e. (SubGrp ` U ) )
62	54, 38	sseldd 3445	. . . . 5 |- ( ph -> B e. (SubGrp ` U ) )
63	54, 49	sseldd 3445	. . . . 5 |- ( ph -> ( L ` C ) e. (SubGrp ` U ) )
64	14	lsmlub 17370	. . . . 5 |- ( ( ( ._|_ ` { X , Y } ) e. (SubGrp ` U ) /\ B e. (SubGrp ` U ) /\ ( L ` C ) e. (SubGrp ` U ) ) -> ( ( ( ._|_ ` { X , Y } ) C_ ( L ` C ) /\ B C_ ( L ` C ) ) <-> ( ( ._|_ ` { X , Y } ) ( LSSum ` U ) B ) C_ ( L ` C ) ) )
65	61, 62, 63, 64	syl3anc 1276	. . . 4 |- ( ph -> ( ( ( ._|_ ` { X , Y } ) C_ ( L ` C ) /\ B C_ ( L ` C ) ) <-> ( ( ._|_ ` { X , Y } ) ( LSSum ` U ) B ) C_ ( L ` C ) ) )
66	44, 52, 65	mpbi2and 937	. . 3 |- ( ph -> ( ( ._|_ ` { X , Y } ) ( LSSum ` U ) B ) C_ ( L ` C ) )
67	15, 66	eqsstr3d 3479	. 2 |- ( ph -> ( ._|_ ` { ( X .+ Y ) } ) C_ ( L ` C ) )
68		eqid 2462	. . 3 |- (LSHyp ` U ) = (LSHyp ` U )
69	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	lcfrlem17 35173	. . . 4 |- ( ph -> ( X .+ Y ) e. ( V \ { .0. } ) )
70	1, 2, 3, 4, 6, 68, 9, 69	dochsnshp 35067	. . 3 |- ( ph -> ( ._|_ ` { ( X .+ Y ) } ) e. (LSHyp ` U ) )
71	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 27, 41, 25, 29, 42	lcfrlem34 35190	. . . 4 |- ( ph -> C =/= ( 0g ` D ) )
72	68, 26, 22, 27, 31, 30, 47	lduallkr3 32774	. . . 4 |- ( ph -> ( ( L ` C ) e. (LSHyp ` U ) <-> C =/= ( 0g ` D ) ) )
73	71, 72	mpbird 240	. . 3 |- ( ph -> ( L ` C ) e. (LSHyp ` U ) )
74	68, 30, 70, 73	lshpcmp 32600	. 2 |- ( ph -> ( ( ._|_ ` { ( X .+ Y ) } ) C_ ( L ` C ) <-> ( ._|_ ` { ( X .+ Y ) } ) = ( L ` C ) ) )
75	67, 74	mpbid 215	1 |- ( ph -> ( ._|_ ` { ( X .+ Y ) } ) = ( L ` C ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   =/= wne 2633   E.wrex 2750   {crab 2753   \ cdif 3413   i^i cin 3415   C_ wss 3416   {csn 3980   {cpr 3982   |-> cmpt 4477   ` cfv 5605   iota_crio 6281  (class class class)co 6320   Basecbs 15176   +g cplusg 15245   .rcmulr 15246  Scalarcsca 15248   .scvsca 15249   0gc0g 15393   -gcsg 16726  SubGrpcsubg 16866   LSSumclsm 17341   invrcinvr 17954   LModclmod 18146   LSubSpclss 18210   LSpanclspn 18249  LSAtomsclsa 32586  LSHypclsh 32587  LFnlclfn 32669  LKerclk 32697  LDualcld 32735   HLchlt 32962   LHypclh 33595   DVecHcdvh 34692   ocHcoch 34961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-riotaBAD 32571

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-of 6563  df-om 6725  df-1st 6825  df-2nd 6826  df-tpos 7004  df-undef 7051  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-n0 10904  df-z 10972  df-uz 11194  df-fz 11820  df-struct 15178  df-ndx 15179  df-slot 15180  df-base 15181  df-sets 15182  df-ress 15183  df-plusg 15258  df-mulr 15259  df-sca 15261  df-vsca 15262  df-0g 15395  df-mre 15547  df-mrc 15548  df-acs 15550  df-preset 16228  df-poset 16246  df-plt 16259  df-lub 16275  df-glb 16276  df-join 16277  df-meet 16278  df-p0 16340  df-p1 16341  df-lat 16347  df-clat 16409  df-mgm 16543  df-sgrp 16582  df-mnd 16592  df-submnd 16638  df-grp 16728  df-minusg 16729  df-sbg 16730  df-subg 16869  df-cntz 17026  df-oppg 17052  df-lsm 17343  df-cmn 17487  df-abl 17488  df-mgp 17779  df-ur 17791  df-ring 17837  df-oppr 17906  df-dvdsr 17924  df-unit 17925  df-invr 17955  df-dvr 17966  df-drng 18032  df-lmod 18148  df-lss 18211  df-lsp 18250  df-lvec 18381  df-lsatoms 32588  df-lshyp 32589  df-lcv 32631  df-lfl 32670  df-lkr 32698  df-ldual 32736  df-oposet 32788  df-ol 32790  df-oml 32791  df-covers 32878  df-ats 32879  df-atl 32910  df-cvlat 32934  df-hlat 32963  df-llines 33109  df-lplanes 33110  df-lvols 33111  df-lines 33112  df-psubsp 33114  df-pmap 33115  df-padd 33407  df-lhyp 33599  df-laut 33600  df-ldil 33715  df-ltrn 33716  df-trl 33771  df-tgrp 34356  df-tendo 34368  df-edring 34370  df-dveca 34616  df-disoa 34643  df-dvech 34693  df-dib 34753  df-dic 34787  df-dih 34843  df-doch 34962  df-djh 35009

This theorem is referenced by:  lcfrlem36  35192